894 
of 2 with the nodal curve of 2° lie now however a little differ- 
ently. The points 7, remain 36-fold for the nodal curve and they 
therefore furnish 4 >< 72 = 288 points of intersection, the 58 torsal 
lines of the 2"d kind give 58, the 6 nodal edges give 3 x 6=18 
other ones; the 4 points S; = 7;* however absorb each of them 40 
points of intersection. Let us namely imagine our figure variable 
and in particular / continuously passing into a complex ray, we then 
see how the 4 points 7;* tend more and more to S;, but at the 
sime time how the 40 points of intersection of / with the nodal curve 
group themselves more and more into 4 groups of 10 in such a 
way that each group is as it were attracted by one of the points 
Si; now each of those 40 points counts for 2, each point 7” for 
20 points of those we looked for; so on the moment that 7,* as 
well as the 10 points of the corresponding group coincide with 
S; this point counts for 40, so the four together for 160 and the 
sum of the four numbers printed in heavy type is again 524. 
19. More considerable are the modifications if finally we now 
assume that / becomes a ray of the congruence; nothing is to be 
noticed at 2%, as / remains a ray of the complex, but the other 
locus becomes a surface 2'*, for which / is only a fivefold line. 
The regulus of before is namely now again replaced by a cone 
|P.|, but the vertex itself 2 now lies on @°, because / is a ray 
of the congruence, thus itself a generatrix. It even appears twice 
as a genera:rix; for the cone cuts @° according to a 4** which has 
now a.o. also a nodal point in P; and to this nodal point the line 
/ corresponds twice. A generatrix of the cone { P|] cuts 2° in: Pi and 
in five other points; so through the corresponding focus on / pass 
five generatrices not coinciding with J, i.e. 7 és a sivefold line. 
To a plane 4 through / a twisted cubic is conjugated containing 
the four vertices of the cones and P, and cutting @° in 13 points 
more; so in a plane 4 lie besides / 13 generatrices, i.e. our surface 
is a @'° of order 18 with a fivefold line l. 
Among the generatrices of the cone [P,] there are two touching 
k* in P; and likewise among the twisted eubies; the foci of the 
former are the points of intersection proper of / with two generatrices 
coinciding with /, the planes conjugated to the latter being the con- 
necting planes; thus two particular torsal planes and pinch points 
(see § 20). 
The line 7; is a generatrix of the cone [P,] and it ents 2° 
besides in these twa points in four more; the corresponding four 
7 
rays s pass through S;= 7;* and lie in the plane /7; whilst the 
